For larger values of \(n\) , qualitative behavior remains the same, although the quantitative variation of \(\Delta m^\pm (t)\)  happens over a larger range now. This is expected since the number of receptors per cluster increases, the total change in methylation level of a cluster also increases. The negative values observed at small \(t\)  also show similar trend, the minimum in \(\Delta m^\pm (t)\)  at small \(t\)  becomes deeper as \(n\)  increases. We also notice that for \(n=10,30\)  the range of variation of \(\Delta m^+(t)\)  and \(\Delta m^-(t)\)  are significantly different, but for \(n=100,200\)  the ranges are not as different for the uphill and downhill runs. Methylation is still slower for the downhill curve, but the values are much closer to the uphill curve. This is because for large \(n\)  the activity fluctuations increase and then adaptation plays a bigger role in the signaling network and ligand free energy becomes less important [1]}, [2]}. The difference in the cell behavior during an uphill and downhill run therefore decreases for large \(n\) . In Appendix  we show this explicitly by plotting the difference between average activity during an uphill and downhill run, and as expected, this difference decreases with \(n\)  for large \(n\) . This explains why the temporal variation of methylation follows a similar course for large \(n\) , irrespective of whether the cell is running uphill or downhill.
